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Microbial predation in a periodically operated chemostat: A global study of the interaction between natural and externally imposed frequencies. (English) Zbl 0729.92522

92D40 Ecology
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[1] Arnol’d, V.I., Geometrical methods in the theory of ordinary differential equations, (1983), Springer New York · Zbl 0569.58018
[2] Arnol’d, V.I., Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funct. anal. appl., 11, 1-10, (1977) · Zbl 0411.58013
[3] Aronson, D.G.; Chory, M.A.; Hall, G.R.; McGehee, R.P., Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study, Commun. math. phys., 83, 303-354, (1982) · Zbl 0499.70034
[4] Aronson, D.G.; McGehee, R.P.; Kevrekidis, I.G.; Aris, R., Entrainment regions for periodically forced oscillators, Phys. rev. A, 33, 2190-2192, (1986)
[5] Bader, F.G.; Tsuchiya, H.M.; Fredrickson, A.G., Grazing of ciliates on blue-Green algae: effects of ciliate encystment and related phenomena, Biotechnol. bioeng., 18, 311-332, (1976)
[6] Bailey, J.E., Periodic operation of chemical reactors-a review, Chem. eng. commun., 1, 111-124, (1973)
[7] Berezovskaya, F.S.; Khibnik, A.I., On the bifurcation of separatrices in the problem of stability loss of auto-oscillations near 1:4 resonance, Prikl. mat. mekh. USSR, 44, 663-667, (1981) · Zbl 0485.58015
[8] Bogdanov, R.I., Versal deformations of a singular point on the plane in the case of zero eigenvalues, Func. anal. appl., 9, 144-145, (1975) · Zbl 0447.58009
[9] Bogdanov, R.I., Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues, Sel. sov. math., 1, 389-421, (1981) · Zbl 0518.58030
[10] Bonomi, A.; Fredrickson, A.G., Protozoan feeding and bacterial wall growth, Biotechnol. bioeng., 18, 239-252, (1976)
[11] Bryant, P.; Jeffries, C., The dynamics of phase locking and points of resonance in a forced magnetic oscillator, Physica, 25D, 196-232, (1987) · Zbl 0654.34035
[12] Bungay, H.R.; Bungay, M.L., Microbial interactions in continuous culture, Adv. appl. microbiol., 10, 269-290, (1968)
[13] Butler, G.J.; Hsu, S.B.; Waltman, P., A mathematical model for the chemostat with periodic washout rate, SIAM J. appl. math., 4, 435-449, (1985) · Zbl 0584.92027
[14] Canale, R.P., Predator-prey relationships in a model for the activated process, Biotechnol. bioeng., 11, 887-907, (1969)
[15] Canale, R.P., An analysis of models describing predator-prey interaction, Biotechnol. bioeng., 12, 353-378, (1970)
[16] Canale, R.P.; Lustig, T.D.; Kehrberger, P.M.; Salo, J.E., Experimental and mathematical modeling studies of protozoan predation on bacteria, Biotechnol. bioeng., 15, 707-728, (1973)
[17] Chenciner, A., Hamiltonian-like phenomena in saddle-node bifurcations of invariant curves for plane diffeomorphisms, (), 7-14
[18] Chenciner, A., Bifurcations des points fixes elliptiques. II. orbites periodiques et ensembles de Cantor invariants, Invent. math., 80, 81-106, (1985) · Zbl 0578.58031
[19] Cunningham, A.; Nisbet, R.M., Transients and oscillations in continuous culture, () · Zbl 0544.92020
[20] Curds, C.R., A computer simulation study of predator-prey relationships in a single stage continuous culture system, Water res., 5, 793-812, (1971)
[21] Curds, C.R., A theoretical study of factors influencing the microbial population dynamics of the activated-sludge process. I, Water res., 7, 1269-1284, (1973)
[22] Davison, B.H.; Stephanopoulos, G., Effect of ph oscillations on a competing mixed culture, Biotechnol. bioeng., 28, 1127-1137, (1986)
[23] Doedel, E.J., AUTO: a program for the bifurcation analysis of autonomous systems, Congressus num., AUTO86 user manual, 30, 265-285, (1986), Pasadena
[24] Drake, J.F.; Jost, J.L.; Fredrickson, A.G.; Tsuchiya, H.M., The food chain, (), 67-94, (NASA Spec. Publ. No. 165)
[25] Drake, J.F.; Tsuchiya, H.M., Predation on Escherichia coli by colpoda steinii, Appl. environ. microbiol., 31, 870-874, (1976)
[26] Ecke, R.E.; Kevrekidis, I.G., Interactions of resonances and global bifurcations in Rayleigh-Bénard convection, Phys. lett. A, 131, 344-352, (1988)
[27] Gambaudo, J.M., Perturbation of a Hopf bifurcation by an external time-periodic forcing, J. diff. equations, 57, 172-199, (1985) · Zbl 0516.34042
[28] Glass, L.; Guevara, M.R.; Belair, J.; Shrier, A., Global bifrucations of a periodically forced biological oscillator, Phys. rev. A, 29, 1348-1357, (1984)
[29] Gogolides, E.; Nicolai, J.-P.; Sawin, H.H., Comparison of experimental measurements and model predictions for radio-frequency ar and SF_{6} discharges, J. vac. sci. technol. A, 7, 1001-1006, (1989)
[30] Gollub, J.P.; Benson, S.V., Many routes to turbulent convection, J. fluid mech., 100, 449-470, (1980)
[31] Graves, D.B.; Jensen, K.F., A continuum model of DC and RF discharges, IEEE trans. plasma sci., PS-14, 78-91, (1986)
[32] Guevara, M.R.; Glass, L.; Shrier, A., Phase locking, period doubling bifurcations and irregular dynamics in periodically stimulated cardiac cells, Science, 24, 1350, (1981)
[33] Hale, J.K.; Somolinos, A.S., Competition for fluctuating nutrient, J. math. biol., 18, 255-280, (1983) · Zbl 0525.92024
[34] Hayashi, C., Non-linear oscillations in physical systems, (1965), McGraw-Hill New York
[35] Hindmarsh, A.C., LSODE and LSODI: two initial-value ordinary differential equations solvers, ACM-signum newlsett., 15, 10-11, (1980)
[36] Holmes, P.J.; Moon, F.C., Strange attractors and chaos in nonlinear mechanics, J. appl. mech., 50, 1021-1032, (1983)
[37] Hsu, S.B., A competition model for a seasonally fluctuating nutrient, J. math. biol., 9, 115-132, (1980) · Zbl 0431.92027
[38] Iooss, G., Bifurcations of maps and applications, (1979), Elsevier/North-Holland New York · Zbl 0408.58019
[39] Jensen, M.H.; Bak, P.; Bohr, T., Complete Devil’s staircase, fractal dimension, and universality of mode-locking structure in the circle map, Phys. rev. lett., 50, 1637-1639, (1983)
[40] Jost, J.L.; Drake, J.F.; Fredrickson, A.G.; Tsuchiya, H.M., Interactions of tetrahymena pyriformis, Escherichia coli, azotobacter vinelandii, and glucose in a minimal medium, J. bacteriol., 113, 834-840, (1973)
[41] Jost, J.L.; Drake, J.F.; Tsuchiya, H.M.; Fredrickson, A.G., Microbial food chains and food webs, J. theor. biol., 41, 461-484, (1973)
[42] Kai, T.; Tomita, K., Stroboscopic phase portrait of a forced nonlinear oscillator, Prog. theor. phys., 61, 54-73, (1979)
[43] Kevrekidis, I.G., A numerical study of global bifurcations in chemical dynamics, Aiche j., 33, 1850-1864, (1987)
[44] I. G. Kevrekidis and M.S. Jolly, On the use of interactive graphics in the numerical study of chemical dynamics, paper no. 22c, presented to the 1987 AIChE Annual Meeting, New York November 1987
[45] Kevrekidis, I.G.; Schmidt, L.D.; Aris, R., The stirred tank forced, Chem. eng. sci., 41, 1549-1560, (1986)
[46] Kevrekidis, I.G.; Schmidt, L.D.; Aris, R.; Pelikan, S., Numerical computation of invariant circles of maps, Physica, 16D, 243-251, (1985) · Zbl 0581.58030
[47] Leis, J.R.; Kramer, M.A., ODESSA—an ordinary differential equation solver with explicit simultaneous sensitivity analysis, ACM trans. math. software, 14, 61-67, (1985) · Zbl 0639.65043
[48] Mackay, R.S.; Tresser, C., Transition to topological chaos for circle maps, Physica, 19D, 206-237, (1986) · Zbl 0596.58027
[49] Matsubara, M.; Watanabe, N.; Hasegawa, S., Bifurcations in a bang-bang controlled mixed culture system, Chem. eng. sci., 41, 523-531, (1986)
[50] McKarnin, M.A.; Schmidt, L.D.; Aris, R., Forced oscillations of a self-oscillating bimolecular surface reaction model, Proc. roy. soc. (lond.) A, 417, 363-388, (1988) · Zbl 0669.34043
[51] Monod, J., Recherches sur la croissance des cultures bacteriennes, (1942), Hermann Paris
[52] Neishtadt, A.I., Bifurcations of the phase pattern of an equation system arising in the problem of stability loss of self-oscillations close to 1:4 resonance, Prikl. mat. mekh. USSR, 42, 896-907, (1979) · Zbl 0419.58013
[53] Nisbet, R.M.; Cunningham, A; Gurney, W.S.C., Endogenous metabolism and the stability of prey-predator systems, Biotechnol. bioeng., 25, 301-306, (1983)
[54] Ostlund, S.; Rand, D.; Sethna, J.; Siggia, E., Universal properties of the transition from quasi-periodicity to chaos in dissipative systems, Physica, 8D, 303-342, (1983) · Zbl 0538.58025
[55] Pavlou, S., Dynamics of a chemostat in which one microbial population feeds on another, Biotechnol. bioeng., 27, 1525-1532, (1985)
[56] Pavlou, S.; Fredrickson, A.G., Effects of the inability of suspension-feeding protozoa to collect all cell sizes of a bacterial population, Biotechnol. bioeng., 25, 1747-1772, (1983)
[57] Pavlou, S.; Kevrekidis, I.G.; Lyberatos, G., On the coexistence of competing microbial species in a chemostat under cycling, Biotechnol. bioeng., 35, 224-232, (1990)
[58] Peckham, B.B., The necessity of Hopf bifurcation in periodically forced oscillators, (), Nonlinearity, 3, 261-380, (1990) · Zbl 0704.58035
[59] Peckham, B.B.; Kevrekidis, I.G., Higher order degeneracies in the local period doubling bifurcation for diffeomorphisms, SIAM J. math. anal., (1991), in press · Zbl 0744.58059
[60] Proper, G.; Garver, J., Mass culture of the protozoa colpoda steinii, Biotechnol. bioeng., 8, 287-296, (1966)
[61] Ratnam, D.A.; Pavlou, S.; Fredrickson, A.G., Effects of attachment of bacteria to chemostat walls in a microbial predator-prey relationship, Biotechnol. bioeng., 24, 2675-2694, (1982)
[62] Sambanis, A.; Pavlou, S.; Fredrickson, A.G., Analysis of the dynamics of ciliate bacterial interactions in a CSTR, Chem. eng. sci., 41, 1455-1469, (1986)
[63] Sambanis, A.; Pavlou, S.; Fredrickson, A.G., Coexistence of bacteria and feeding ciliates: growth of bacteria on autochtonous substrates as a stabilizing factor for coexistence, Biotechnol. bioeng., 29, 714-728, (1987)
[64] Schreiber, I.; Dolnik, M.; Choc, P.; Marek, M., Resonance behaviour in two-parameter families of periodically forced oscillators, Phys. lett. A, 128, 66-70, (1988)
[65] Smith, H.L., Competitive coexistence in an oscillating chemostat, SIAM J. appl. math., 40, 498-522, (1981) · Zbl 0467.92018
[66] Stavans, J.; Heslot, F.; Libchaber, A., Fixed winding number and the quasiperiodic route to chaos in a convective fluid, Phys. rev. lett., 55, 596-599, (1985)
[67] Stephanopoulos, G.; Fredrickson, A.G.; Aris, R., The growth of competing microbial populations in a CSTR with periodically varying inputs, Aiche j., 25, 863-872, (1979)
[68] Stephens, M.L.; Lyberatos, G., Effect of cycling on final mixed culture fate, Biotechnol. bioeng., 29, 672-678, (1987)
[69] Sudo, R.; Kobayashi, K.; Aiba, S., Some experiments and analysis of a predator-prey model interaction between colpidium campylum and alcaligenes faecalis in continuous and mixe culture, Biotechnol. bioeng., 17, 167-184, (1975)
[70] Takens, F., Singularities of vector fields, Publ. math. IHES, 43, 47-100, (1974) · Zbl 0279.58009
[71] Thoulouze-Pratt, E., Numerical analysis of the behaviour of an almost periodic solution to a periodic differential equation, an example of successive bifurcations of invar iant tori, Lect. notes biomath., 49, 265-271, (1983)
[72] Tsuchiya, H.M.; Drake, J.F.; Jost, J.L.; Fredrickson, A.G., Predator-prey interactions of dictyostelium discoideum and Escherichia coli in continuous culture, J. bacteriol., 110, 1147-1153, (1972)
[73] Ueda, Y., Steady motions exhibited by Duffing’s equation: a picture book of regular and chaotic motions, (), 311-322
[74] Vance, W.; Ross, J., A detailed study of a forced chemical oscillator: arnol’d tongues and bifurcation sets, J. chem. phys., 91, 7654-7670, (1989)
[75] van den Ende, P., Predator-prey interactions in continuous culture, Science, 181, 562-564, (1973)
[76] Van Veldhuizen, M., A new algorithm for the numerical approximation of an invariant curve, SIAM J. sci. stat. comp., 8, 951-962, (1987) · Zbl 0657.65098
[77] Villarreal, E.; Canale, R.P.; Akcasu, I., A multigroup model for predator-prey interactions, Biotechnol. bioeng., 17, 1269-1290, (1975)
[78] Westervelt, R.M.; Teitsworth, S.W., Nonlinear dynamics and chaos in extrinsic photoconductors, Physica, 23D, 187-194, (1986)
[79] Williams, F.M., On understanding predator-prey interactions, ()
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